Two identical 19.8-kg spheres of radius 12 cm are 29.2 cm apart (center-to-center distance).
(a) If they are released from rest and allowed to fall toward one another, what is their speed when they first make contact?
(b) If the spheres are initially at rest and just touching, how much energy is required to separate them to 1.03 m apart? Assume that the only force acting on each mass is the gravitational force due to the other mass.
a. put a coordinate system where one is fixed. The potential energy of the other is GMM (1/.24-1/.29) that is PE of the starting position minus the final position (24cm cneter to center).
That PE turns to KE (1/2 M V2),but in reality, both are moving, so the KE is divided between the two, so 1/2 MV^2=1/2 PE
so you can find then the V each has when colliding.
Thanks...I figured out part b
To answer both parts of the question, we need to apply principles from Newton's laws of motion and the laws of gravitation.
(a) To find the speed at which the spheres first make contact, we can use the principle of conservation of mechanical energy.
The initial total mechanical energy of the system, when the spheres are released from rest, consists of both the kinetic energy and gravitational potential energy.
The gravitational potential energy of each sphere is given by:
PE = m * g * h,
where m is the mass of each sphere, g is the acceleration due to gravity, and h is the height from which it falls.
Let's calculate the gravitational potential energy of one sphere:
PE = (19.8 kg) * (9.8 m/s^2) * (0.12 m) = 23.208 J.
Since there are two spheres, the initial total gravitational potential energy is:
PE_total = 2 * 23.208 J = 46.416 J.
At the point of contact, the spheres have fallen a distance equal to their original separation (29.2 cm = 0.292 m). At that point, all of the initial potential energy would have been converted to kinetic energy.
Therefore, the initial total kinetic energy is:
KE_total = PE_total = 46.416 J.
Since both spheres have identical masses, the kinetic energy will be distributed equally between them.
KE1 = KE2 = KE_total / 2 = 46.416 J / 2 = 23.208 J.
The kinetic energy is given by:
KE = (1/2) * m * v^2,
where v is the speed of each sphere when they first make contact.
Plugging in the values, we can solve for v:
23.208 J = (1/2) * (19.8 kg) * v^2.
Simplifying the equation and solving for v, we get:
v = sqrt( (23.208 J * 2) / (19.8 kg) ) = 1.94 m/s.
Therefore, the speed when the spheres first make contact is approximately 1.94 m/s.
(b) To determine the amount of energy required to separate the spheres to a distance of 1.03 m apart, we need to calculate the change in gravitational potential energy.
The final gravitational potential energy is given by:
PE_final = m * g * h_final,
where h_final is the final height, which is equal to the separation distance between the spheres.
The initial gravitational potential energy was calculated in part (a) as 46.416 J, and the final potential energy is:
PE_final = (19.8 kg) * (9.8 m/s^2) * (1.03 m) = 200.1186 J (rounded to 4 decimal places).
The change in gravitational potential energy (ΔPE) is given by:
ΔPE = PE_final - PE_initial,
ΔPE = 200.1186 J - 46.416 J = 153.7026 J (rounded to 4 decimal places).
Therefore, it would require approximately 153.7026 J of energy to separate the spheres to a distance of 1.03 m apart, assuming that the only force acting on each mass is the gravitational force due to the other mass.